Strong Solutions for the Stochastic Cahn-Hilliard Convective Brinkman-Forchheimer Model for Tumor Growth

Abstract

In this work, we analyze a diffuse-interface model for tumor growth, subject to multiplicative white noises, posed on a bounded domain $\mathcal{O} \subset \mathbb{R}^d$, $d=2,3$. The model couples a stochastic incompressible convective Brinkman-Forchheimer (CBF) equation or Navier-Stokes equation with damping $\eta|v|^{r-1}v $ for the averaged velocity field $v$, to a Cahn-Hilliard (CH) equation for the phase field variable $\phi$ and to a stochastic reaction-diffusion equation governing the nutrient concentration $\sigma$. We establish the existence of local strong solutions , for $ r \geq 1 $ in $d=2$ and $ r \in [1,3] $ in $d=3$. We prove the weak-strong uniqueness holds in both $d = 2, 3$. In addition, for $d = 2$, the uniqueness of weak solutions is obtained for all $\eta,\nu > 0$, and $r \geq 1$, while it holds in $d = 3$ for $r \geq 3$ and $\eta \nu \geq 1$ when $r = 3$ under an assumption on $\sigma$. Moreover, for $d=2$ and $r \in [1,3]$, we obtain that the strong solution exists globally in time.